#### 5.1. The Binary Take-up Model

We use two different estimators of the binary take-up model: the probit technique which assumes normality; and the semi-parametric estimator of Klein and Spady (KS) (1993), which, in its simplest form, maximises the following quasi-log-likelihood:

- (10)

where summations are over the set of *n* pensioner households for whom *B*_{i} > 0, τ_{i} is the dependent variable equal to 1 for IS participation and 0 for non-participation, and is a non-parametric kernel estimate of the regression function of τ_{i} on ln*B*_{i} − **Z**_{i}**α**. We use the Gaussian kernel:

- (11)

where ϕ(.) is the standard normal density function. Note that is not normalised to have zero mean and unit variance. Scale and location are normalised by fixing the coefficient of ln*B*_{i} at unity and excluding the intercept term from the linear form **Zα**. This choice of normalisation does not affect the construction of implicit cost estimates. We experimented with fixed and adaptive bandwidths (the latter using the Breiman *et al.*, 1977, method), with remarkably little effect on the results; those reported here are based on a fixed bandwidth *h* = 0.6.

Table II gives estimates of the stigma/claim cost coefficients **α**. Precise definitions and summaries of the variables are given by Hernandez *et al.* (2006). For the probit model the estimates are calculated as minus the coefficients of the relevant variables divided by the coefficient of ln*B*_{i}. The estimates are the outcome of an extensive process of specification search. The chosen form is superior (in likelihood terms) to other models with alternative functional forms for income and entitlement. There has been some attention paid by sociologists to neighbourhood influences on welfare participation behaviour, with the conclusion that high local rates of poverty, welfare dependency and density of population lead to higher rates of take-up, because of lower social stigma and better local information and support, reducing claim costs (Hirschl and Rank, 1999). We are only able to match survey respondents to large regions rather than neighbourhoods and there are, consequently, no locational effects detectable. We are also able to accept our specification against models with fuller demographic structure and a more general specification involving the ages and education levels of both members for two-person households. Annual dummy variables were also included and found to be insignificant with χ^{2} statistics of 3.33 and 5.50 for samples 1 and 2, respectively, and a 5% critical value of 9.488. To guard against pre-test bias, we have worked from ‘general’ models to ‘specific’ ones, using a conservative criterion, retaining explanatory variables with asymptotic *t*-ratios in excess of 1.0. Besides log entitlement, the main factors generating high claim costs emerge as income per head, education, status as a recipient of disability benefit, owner-occupation and newly entitled status.

The estimated effect of income is always significant at the 5% level but varies considerably over the two samples. For the probit model estimated on sample 1 data, the coefficient implies a large 11% increase in expected claim costs for each additional £1 of original income. This falls to just over 4% when the data from sample 2 are used. For the KS estimates the range is similar: a 13% impact on sample 1 data but under 5% for sample 2 data. Housing tenure is closely linked to social status as well as wealth. Being a home-owner greatly increases the barriers to IS take-up, increasing estimated mean claim costs as much as eightfold. This is likely to reflect the relatively poor access that home-owners have to information and advice, which is available (through housing associations and local authority housing offices) to most renters.6

Education has a large effect. Having schooling past age 14 is estimated to quadruple expected claim costs on the basis of models estimated from sample 1 data. The expected claim costs are lower when sample 2 is used; nevertheless, having schooling past the age of 14 more than doubles claim costs. Although better-educated people may have greater capacity to negotiate the intricacies of the benefit system, on this evidence they must also typically be more vulnerable to stigma or tend to be in circumstances entailing greater costs of claiming. The two disability variables reflect the household's status as a recipient of a (medically assessed but non-means-tested) disability benefit or as one containing a registered disabled person. These have respectively positive and negative impacts on expected claim costs. Note that registering as a disabled person is voluntary and has no direct implications for benefit entitlement, but may bring other benefits such as unrestricted car parking. These two variables summarise a combination of factors. One might interpret the coefficient of the former variable as an indicator of physical impairment which increases the physical difficulty of coping with the IS claims process and thus increases implicit claim costs (roughly twofold). The latter might be interpreted as an indicator of low vulnerability to stigma: those who are willing to seek formal recognition of disability may also tend to be more willing to accept an IS-dependent status and thus have lower expected claim costs (by around 58%). In the absence of direct information on physical capacity, such interpretations are necessarily speculative.

The probit and KS estimates have similar qualitative implications in the samples considered here. However, there is an important difference for age. Using the probit model, claim costs are estimated to increase with age, although at a decreasing rate. If accepted, this result would be hard to rationalise. It seems unlikely that people who claim benefit when younger would cease to do so when they reach a critical age. If the age effect arises through the acquisition of information through time, one would expect the take-up rate to be increasing. Adjustment models based on random durations of periods of need (see Anderson and Meyer, 1997) are inappropriate here and again suggest rising take-up rates. Another interpretation would be that the age variable reflects a cohort effect implying a gradual upward drift in take-up rates over time but this conflicts with the absence of a trend in IS take-up among pensioners at the macro level (DWP, 2004a, and earlier). The issue is resolved once the more flexible semi-parametric approach is used, since age becomes insignificant for all samples. This last finding provides a good illustration of the often-neglected proposition that misspecification of distributional form can cause serious biases in binary response models.

Finally, we have included in the model a dummy variable to reflect the possibility that there is some delay in adjusting to changes in the rules of the benefit system. In April 2001 there was a major revision in the IS rules, which significantly extended entitlement. A dummy variable is used to identify cases of individuals who are entitled in the year they are interviewed but who would have been non-entitled (with unchanged circumstances) under the previous year's rules. This variable has a strongly significant coefficient, implying the existence of adjustment lags.

Figure 2 shows the distribution functions (.) for the probit and KS models in sample 2. To make these comparable, the probit probability, Φ(.), is plotted against the standardised KS estimate. The most striking difference between the two distributions is the fatter upper tail of the KS estimate and a local concentration at around − 1 standard deviations in the lower tail.

#### 5.2. Estimates of the Implicit Stigma/Claim Costs

Table III shows estimated claim costs for the subsample of pensioners receiving IS, constructed using (6) and (9). Means and medians differ substantially but the semi-parametric estimates give much higher estimated claim costs than the probit model, whatever method is used to construct implicit costs. Results are sensitive to the choice of sample, with larger costs estimated in sample 2, where recorded rather than simulated benefit receipt is used when possible. For the preferred KS estimates, the average estimated claim cost for IS recipients is around £4 per week in sample 2 and £3.40 in sample 1 (15% and 13% of mean entitlement, respectively).

Figure 3 shows the empirical distribution of these estimated claim costs for the subset of pensioners in sample 2 observed receiving IS. The KS estimates imply greater dispersion, especially using the conditional mean method to construct implicit costs.

How do these estimates compare with others in the literature? There are no directly comparable figures available, since other researchers have not taken account of the conditioning on observed take-up. For example, Duclos (1995, p. 409) gives expected costs of claiming Supplementary Benefits (SB) in Britain in 1985. Among the pensioner cases, take-up costs range from over £3 per week for single pensioners to over £20 for couples. However, these are not conditioned on the take-up event. The analysis closest to our own is the work on Housing Benefit (HB) by Blundell *et al.* (1988), who estimated claim costs by finding the level of entitlement at which the unconditional take-up probability is 0.5. From the published probit coefficients and sample means relating to retired/unoccupied respondents (Blundell *et al.*, 1988, pp. 73–74), we can apply (7) and (9) to estimate implicit claim costs for the average 1984 HB-entitled pensioner. These are £1.79 and £1.08 (updated to 2002 prices) using the conditional mean and median methods. These are lower than our 1997–2002 estimates for IS, which has a lower take-up rate and, presumably, higher claim costs than HB.

The claim costs faced by those who do not participate in the IS programme cannot be estimated reliably. For participants, claim costs are bounded by the amount of entitlement *B*, but for non-participants they are unbounded. The conditional mean method is particularly unstable since it is heavily influenced by the tail behaviour of the function *F*(.), which is not well-determined statistically. Good estimates of the upper tail of the claim costs distribution would require sample cases with large entitlements, but this is prevented by the design of the benefit system.